Question: Multiply the following complex numbers, marked as blue dots on the graph: $[\cos(\frac{1}{3}\pi) + i \sin(\frac{1}{3}\pi)] \cdot [7(\cos(\pi) + i \sin(\pi))]$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $\cos(\frac{1}{3}\pi) + i \sin(\frac{1}{3}\pi)$ ) has angle $\frac{1}{3}\pi$ and radius $1$ The second number ( $7(\cos(\pi) + i \sin(\pi))$ ) has angle $\pi$ and radius $7$ The radius of the result will be $1 \cdot 7$ , which is $7$ The angle of the result is $\frac{1}{3}\pi + \pi = \frac{4}{3}\pi$ The radius of the result is $7$ and the angle of the result is $\frac{4}{3}\pi$.